Circle of apollonius equation...Please help asap!

[dmp98]

Two fixed points [FONT=MathJax_Math]A[/FONT] and [FONT=MathJax_Math]B[/FONT] lie in the plane, and the distance between them is [FONT=MathJax_Math]A[FONT=MathJax_Math]B[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Math]a[/FONT][/FONT], where [FONT=MathJax_Math]a[FONT=MathJax_Main]>[/FONT][FONT=MathJax_Main]0[/FONT][/FONT].

A point [FONT=MathJax_Math]P[/FONT] moves in the plane so that the ratio of its distances from [FONT=MathJax_Math]A[/FONT] and [FONT=MathJax_Math]B[/FONT] is constant:

[FONT=MathJax_Math]P​

[FONT=MathJax_Math]A[/FONT][FONT=MathJax_Math]P[/FONT][FONT=MathJax_Math]B[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]λ[/FONT][FONT=MathJax_Main],[/FONT]​

[/FONT]​

where [FONT=MathJax_Math]λ[FONT=MathJax_Main]>[/FONT][FONT=MathJax_Main]0[/FONT][/FONT].

1. Can you sketch the locus of the point [FONT=MathJax_Math]P[/FONT] for different values of [FONT=MathJax_Math]λ[/FONT]?
2. Using Cartesian coordinates, work out (the equation of) the locus of [FONT=MathJax_Math]P[/FONT].

[h=4]Suggestion[/h]
You may find it more straightforward to first work with specific values of [FONT=MathJax_Math]a[/FONT] and [FONT=MathJax_Math]λ[/FONT], say [FONT=MathJax_Math]a[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]2[/FONT][/FONT] and [FONT=MathJax_Math]λ[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]3[/FONT][/FONT].

I really need help with finding the equation. I have set point A(xa,ya), B(ya,yb) and P(x,y)
I think rearranging them and putting them into an expresion and then simplifying will probably give the equation but i just don't know how i would rearrange to get one final equation.

Any help or ideas will be higly appreciated as question due tomorrow :O

 

[Deleted member 4993]

Two fixed points [FONT=MathJax_Math]A[/FONT] and [FONT=MathJax_Math]B[/FONT] lie in the plane, and the distance between them is [FONT=MathJax_Math]A[FONT=MathJax_Math]B[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Math]a[/FONT][/FONT], where [FONT=MathJax_Math]a[FONT=MathJax_Main]>[/FONT][FONT=MathJax_Main]0[/FONT][/FONT].

A point [FONT=MathJax_Math]P[/FONT] moves in the plane so that the ratio of its distances from [FONT=MathJax_Math]A[/FONT] and [FONT=MathJax_Math]B[/FONT] is constant:

[FONT=MathJax_Math]P​

[FONT=MathJax_Math]A[/FONT][FONT=MathJax_Math]P[/FONT][FONT=MathJax_Math]B[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]λ[/FONT][FONT=MathJax_Main],[/FONT]​

[/FONT]​

where [FONT=MathJax_Math]λ[FONT=MathJax_Main]>[/FONT][FONT=MathJax_Main]0[/FONT][/FONT].

1. Can you sketch the locus of the point [FONT=MathJax_Math]P[/FONT] for different values of [FONT=MathJax_Math]λ[/FONT]?
2. Using Cartesian coordinates, work out (the equation of) the locus of [FONT=MathJax_Math]P[/FONT].

Suggestion


You may find it more straightforward to first work with specific values of [FONT=MathJax_Math]a[/FONT] and [FONT=MathJax_Math]λ[/FONT], say [FONT=MathJax_Math]a[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]2[/FONT][/FONT] and [FONT=MathJax_Math]λ[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]3[/FONT][/FONT].

I really need help with finding the equation. I have set point A(xa,ya), B(ya,yb) and P(x,y)
I think rearranging them and putting them into an expresion and then simplifying will probably give the equation but i just don't know how i would rearrange to get one final equation.

Any help or ideas will be higly appreciated as question due tomorrow :O

As suggested, did you work with specific values of [FONT=MathJax_Math]a[/FONT] and [FONT=MathJax_Math]λ[/FONT], say [FONT=MathJax_Math]a[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]2[/FONT][/FONT] and [FONT=MathJax_Math]λ[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]3[/FONT][/FONT]? What did you get?

What are your thoughts?

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